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In this paper, we determine the achromatic and diachromatic numbers of somecirculant graphs and digraphs each one with two lengths and give bounds forother circulant graphs and digraphs with two lengths. In particular, for theachromatic number we state that $\alpha(C_{16q^2+20q+7}(1,2))=8q+5$, and forthe diachromatic number we state that$dac(\overrightarrow{C}_{32q^2+24q+5}(1,2))=8q+3$. In general, we give thelower bounds $\alpha(C_{4q^2+aq+1}(1,a))\geq 4q+1$ and$dac(\overrightarrow{C}_{8q^2+2(a+4)q+a+3}(1,a))\geq 4q+3$ when $a$ is a nonquadratic residue of $\mathbb{Z}_{4q+1}$ for graphs and $\mathbb{Z}_{4q+3}$ fordigraphs, and the equality is attained, in both cases, for $a=3$. Finally, wedetermine the achromatic index for circulant graphs of $q^2+q+1$ vertices whenthe projective cyclic plane of odd order $q$ exists.

Achromatic number, achromatic index and diachromatic number of circulant graphs and digraphs